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From continuum excitations to sharp magnons via transverse magnetic field in the spin-12 Ising-like triangular lattice antiferromagnet Na2BaCo(PO4)2

Leonie Woodland1,2, Ryutaro Okuma1,3, J. Ross Stewart2, Christian Balz2,*, and Radu Coldea1

  • 1Clarendon Laboratory, University of Oxford, Physics Department, Parks Road, Oxford OX1 3PU, United Kingdom
  • 2ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, United Kingdom
  • 3Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan

  • *Present address: Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA.
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Phys. Rev. B 112, 104413 – Published 9 September, 2025

DOI: https://doi.org/10.1103/1pvl-kzjm

Abstract

We report high-resolution inelastic neutron scattering measurements of the excitation spectrum in large single crystals of the spin-1/2 triangular-lattice Ising-like antiferromagnet Na2BaCo(PO4)2 in magnetic fields applied transverse to the Ising axis. In the high-field polarized phase above a critical field BC, we observe sharp magnons, as expected in the case of no exchange disorder. Through simultaneous fits to the dispersions including data in a polarizing field along the Ising axis, we obtain an excellent match to an Ising-like XXZ Hamiltonian and rule out previously proposed Kitaev exchanges. In the intermediate-field phase below BC, we observe three dispersive modes, out of which only the lowest energy one is sharp and the others are broad and overlap with continuum scattering. We propose that the broadening effects are due to magnon decays into two-magnon excitations and confirm that such processes are kinematically allowed. The continuum scattering becomes progressively stronger upon lowering the field and, at 0.25 T and zero field, it dominates the entire spectrum with no clear evidence for even broadened magnon modes. We discuss the relevance of the continuous manifold of mean-field degenerate ground states of the refined Hamiltonian for capturing the observed spectrum in zero field, and compare the data with the one- and two-magnon spectrum averaged over this manifold. We also propose a model of the interlayer couplings to explain the observed finite interlayer magnetic propagation vector of the zero-field magnetic order; this requires the breaking of the mirror symmetry in the nominal P3¯m1 space group and through refinement of x-ray diffraction data on an untwinned single crystal, we indeed confirm a rotation of the CoO6 octahedra around the c axis, which lowers the symmetry to P3¯.

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References (55)

  1. R. R. P. Singh and D. A. Huse, Three-sublattice order in triangular- and kagomé-lattice spin-half antiferromagnets, Phys. Rev. Lett. 68, 1766 (1992).
  2. B. Bernu, P. Lecheminant, C. Lhuillier, and L. Pierre, Exact spectra, spin susceptibilities, and order parameter of the quantum Heisenberg antiferromagnet on the triangular lattice, Phys. Rev. B 50, 10048 (1994).
  3. L. Capriotti, A. E. Trumper, and S. Sorella, Long-range Néel order in the triangular Heisenberg model, Phys. Rev. Lett. 82, 3899 (1999).
  4. G. H. Wannier, Antiferromagnetism. the triangular Ising net, Phys. Rev. 79, 357 (1950).
  5. P. Fazekas and P. W. Anderson, On the ground state properties of the anisotropic triangular antiferromagnet, Philos. Mag. 30, 423 (1974).
  6. B. Kleine, E. Müller-Hartmann, K. Frahm, and P. Fazekas, Spin-wave analysis of easy-axis quantum antiferromagnets on the triangular lattice, Z. Phys. B: Condens. Matter 87, 103 (1992).
  7. M. Ulaga, J. Kokalj, A. Wietek, A. Zorko, and P. Prelovšek, Finite-temperature properties of the easy-axis Heisenberg model on frustrated lattices, Phys. Rev. B 109, 035110 (2024).
  8. C. A. Gallegos, S. Jiang, S. R. White, and A. L. Chernyshev, Phase diagram of the easy-axis triangular-lattice J1−J2 model, Phys. Rev. Lett. 134, 196702 (2025).
  9. D. Yamamoto, G. Marmorini, M. Tabata, K. Sakakura, and I. Danshita, Magnetism driven by the interplay of fluctuations and frustration in the easy-axis triangular XXZ model with transverse fields, Phys. Rev. B 100, 140410(R) (2019).
  10. Y. Xu, J. Hasik, B. Ponsioen, and A. H. Nevidomskyy, Simulating spin dynamics of supersolid states in a quantum Ising magnet, Phys. Rev. B 111, L060402 (2025).
  11. R. Zhong, S. Guo, G. Xu, Z. Xu, and R. J. Cava, Strong quantum fluctuations in a quantum spin liquid candidate with a Co-based triangular lattice, Proc. Natl. Acad. Sci. USA 116, 14505 (2019).
  12. N. Li, Q. Huang, X. Y. Yue, W. J. Chu, Q. Chen, E. S. Choi, X. Zhao, H. D. Zhou, and X. F. Sun, Possible itinerant excitations and quantum spin state transitions in the effective spin-1/2 triangular-lattice antiferromagnet Na2BaCo(PO4)2, Nat. Commun. 11, 4216 (2020).
  13. Y. Gao, Y.-C. Fan, H. Li, F. Yang, X.-T. Zeng, X.-L. Sheng, R. Zhong, Y. Qi, Y. Wan, and W. Li, Spin supersolidity in nearly ideal easy-axis triangular quantum antiferromagnet Na2BaCo(PO4)2, npj Quantum Mater. 7, 89 (2022).
  14. J. Sheng, L. Wang, A. Candini, W. Jiang, L. Huang, B. Xi, J. Zhao, H. Ge, N. Zhao, Y. Fu et al., Two-dimensional quantum universality in the spin-1/2 triangular-lattice quantum antiferromagnet Na2BaCo(PO4)2, Proc. Natl. Acad. Sci. USA 119, e2211193119 (2022).
  15. B. S. Mou, X. Zhang, L. Xiang, Y. Xu, R. Zhong, R. J. Cava, H. Zhou, Z. Jiang, D. Smirnov, N. Drichko, and S. M. Winter, Comparative Raman scattering study of crystal field excitations in Co-based quantum magnets, Phys. Rev. Mater. 8, 084408 (2024).
  16. J. Xiang, C. Zhang, Y. Gao, W. Schmidt, K. Schmalzl, C.-W. Wang, B. Li, N. Xi, X.-Y. Liu, H. Jin et al., Giant magnetocaloric effect in spin supersolid candidate Na2BaCo(PO4)2, Nature (London) 625, 270 (2024).
  17. R. Coldea, D. A. Tennant, K. Habicht, P. Smeibidl, C. Wolters, and Z. Tylczynski, Direct measurement of the spin Hamiltonian and observation of condensation of magnons in the 2D frustrated quantum magnet Cs2CuCl4, Phys. Rev. Lett. 88, 137203 (2002).
  18. M. Zhu, V. Romerio, N. Steiger, S. D. Nabi, N. Murai, S. Ohira-Kawamura, K. Y. Povarov, Y. Skourski, R. Sibille, L. Keller, Z. Yan, S. Gvasaliya, and A. Zheludev, Continuum excitations in a spin supersolid on a triangular lattice, Phys. Rev. Lett. 133, 186704 (2024).
  19. T. Chen, A. Ghasemi, J. Zhang, L. Shi, Z. Tagay, L. Chen, E.-S. Choi, M. Jaime, M. Lee, Y. Hao et al., Phase diagram and spectroscopic evidence of supersolids in quantum Ising magnet K2Co(SeO3)2, arXiV:2402.15869.
  20. Y. Y. Huang, D. Z. Dai, C. C. Zhao, J. M. Ni, L. S. Wang, B. L. Pan, B. Gao, P. Dai, and S. Y. Li, Thermal conductivity of triangular-lattice antiferromagnet Na2BaCo(PO4)2: Absence of itinerant fermionic excitations, arXiv:2206.08866.
  21. X.-Y. Liu, Y. Gao, H. Li, W. Jin, J. Xiang, H. Jin, Z. Chen, W. Li, and G. Su, Quantum spin liquid candidate as superior refrigerant in cascade demagnetization cooling, Commun. Phys. 5, 233 (2022).
  22. X. Xu, Z. Wu, Y. Chen, Q. Huang, Z. Hu, X. Shi, K. Du, S. Li, R. Bian, R. Yu, Y. Cui, H. Zhou, and W. Yu, NMR study of supersolid phases in the triangular-lattice antiferromagnet Na2BaCo(PO4)2, arXiv:2504.08570.
  23. J. Sheng, L. Wang, W. Jiang, H. Ge, N. Zhao, T. Li, M. Kofu, D. Yu, W. Zhu, J.-W. Mei, Z. Wang, and L. Wu, Continuum of spin excitations in an ordered magnet, Innovation 6, 100769 (2025).
  24. T. I. Popescu, N. Gora, F. Demmel, Z. Xu, R. Zhong, T. J. Williams, R. J. Cava, G. Xu, and C. Stock, Zeeman split Kramers doublets in spin-supersolid candidate Na2BaCo(PO4)2, Phys. Rev. Lett. 134, 136703 (2025).
  25. S. Lee, C. H. Lee, A. Berlie, A. D. Hillier, D. T. Adroja, R. Zhong, R. J. Cava, Z. H. Jang, and K.-Y. Choi, Temporal and field evolution of spin excitations in the disorder-free triangular antiferromagnet Na2BaCo(PO4)2, Phys. Rev. B 103, 024413 (2021).
  26. Y. Gao, C. Zhang, J. Xiang, D. Yu, X. Lu, P. Sun, W. Jin, G. Su, and W. Li, Double magnon-roton excitations in the triangular-lattice spin supersolid, Phys. Rev. B 110, 214408 (2024).
  27. D. Macdougal, S. Williams, D. Prabhakaran, R. I. Bewley, D. J. Voneshen, and R. Coldea, Avoided quasiparticle decay and enhanced excitation continuum in the spin-12 near-Heisenberg triangular antiferromagnet Ba3CoSb2O9, Phys. Rev. B 102, 064421 (2020).
  28. R. Verresen, R. Moessner, and F. Pollmann, Avoided quasiparticle decay from strong quantum interactions, Nat. Phys. 15, 750 (2019).
  29. R. I. Bewley, J. W. Taylor, and S. M. Bennington, LET, a cold neutron multi-disk chopper spectrometer at ISIS, Nucl. Instrum. Methods Phys. Res., Sect. A 637, 128 (2011).
  30. O. Arnold, J. C. Bilheux, J. M. Borreguero, A. Buts, S. I. Campbell, L. Chapon, M. Doucet, N. Draper, R. Ferraz Leal, M. A. Gigg, , Mantid—data analysis and visualization package for neutron scattering and μsr experiments, Nucl. Instrum. Methods Phys. Res., Sect. A 764, 156 (2014).
  31. R. A. Ewings, A. Buts, M. D. Le, J. van Duijn, I. Bustinduy, and T. G. Perring, Horace: Software for the analysis of data from single crystal spectroscopy experiments at time-of-flight neutron instruments, Nucl. Instrum. Methods Phys. Res., Sect. A 834, 132 (2016).
  32. R. Coldea, MSlice: Matlab visualization software for single crystal and powder time-of-flight neutron data (2000), http://mslice.isis.rl.ac.uk/Main_Page.
  33. C. A. Gallegos and A. L. Chernyshev, Magnon interactions in the quantum paramagnetic phase of CoNb2O6, Phys. Rev. B 109, 014424 (2024).
  34. S. Miyashita and H. Kawamura, Phase transitions of anisotropic Heisenberg antiferromagnets on the triangular lattice, J. Phys. Soc. Jpn. 54, 3385 (1985).
  35. A. L. Chernyshev and M. E. Zhitomirsky, Spin waves in a triangular lattice antiferromagnet: Decays, spectrum renormalization, and singularities, Phys. Rev. B 79, 144416 (2009).
  36. M. E. Zhitomirsky and A. L. Chernyshev, Colloquium: Spontaneous magnon decays, Rev. Mod. Phys. 85, 219 (2013).
  37. S. Toth and B. Lake, Linear spin wave theory for single-Q incommensurate magnetic structures, J. Phys.: Condens. Matter 27, 166002 (2015).
  38. B. Gao, T. Chen, C. Liu, M. L. Klemm, S. Zhang, Z. Ma, X. Xu, C. Won, G. T. McCandless, N. Murai et al., Spin excitation continuum in the exactly solvable triangular-lattice spin liquid CeMgAl11O19, arXiv:2408.15957.
  39. R. Chi, J. Hu, H.-J. Liao, and T. Xiang, Dynamical spectra of spin supersolid states in triangular antiferromagnets, Phys. Rev. B 110, L180404 (2024).
  40. A. J. Hearmon, F. Fabrizi, L. C. Chapon, R. D. Johnson, D. Prabhakaran, S. V. Streltsov, P. J. Brown, and P. G. Radaelli, Electric field control of the magnetic chiralities in ferroaxial multiferroic RbFe(MoO4)2, Phys. Rev. Lett. 108, 237201 (2012).
  41. C. Zhang, J. Xiang, C. Su, D. Sheptyakov, X. Liu, Y. Gao, P. Sun, W. Li, G. Su, and W. Jin, Successive magnetic transitions in the spin-5/2 easy-axis triangular-lattice antiferromagnet Na2BaMn(PO4)2: A neutron diffraction study, Phys. Rev. B 110, 214405 (2024).
  42. J. Sheng, J.-W. Mei, L. Wang, X. Xu, W. Jiang, L. Xu, H. Ge, N. Zhao, T. Li, A. Candini et al., Bose–Einstein condensation of a two-magnon bound state in a spin-1 triangular lattice, Nat. Mater. 24, 544 (2025).
  43. Y. Kajita, T. Nagai, S. Yamagishi, K. Kimura, M. Hagihala, and T. Kimura, Ferroaxial transitions in glaserite-type Na2BaM(PO4)2 (M = Mg, Mn, Co, and Ni), Chem. Mater. 36, 7451 (2024).
  44. R. Coldea et al., ISIS Pulsed Neutron and Muon Source (2023), https://doi.org/10.5286/ISIS.E.RB2220680.
  45. R. Coldea et al., ISIS Pulsed Neutron and Muon Source (2024), https://doi.org/10.5286/ISIS.E.RB2320303.
  46. K. Momma and F. Izumi, Vesta3 for three-dimensional visualization of crystal, volumetric and morphology data, J. Appl. Crystallogr. 44, 1272 (2011).
  47. L. Woodland, R. Okuma, J. R. Stewart, C. Balz, and R. Coldea, Oxford University Research Archive (2025), https://dx.doi.org/10.5287/ora-qrd6mzojz.
  48. Agilent, CrysAlis PRO (2014), Agilent Technologies Ltd, Yarnton, Oxfordshire, England.
  49. O. V. Dolomanov, L. J. Bourhis, R. J. Gildea, J. A. K. Howard, and H. Puschmann, Olex2: A complete structure solution, refinement and analysis program, J. Appl. Crystallogr. 42, 339 (2009).
  50. C. Wellm, W. Roscher, J. Zeisner, A. Alfonsov, R. Zhong, R. J. Cava, A. Savoyant, R. Hayn, J. van den Brink, B. Büchner, O. Janson, and V. Kataev, Frustration enhanced by Kitaev exchange in a j=1/2 triangular antiferromagnet, Phys. Rev. B 104, L100420 (2021).
  51. R. Coldea, D. A. Tennant, and Z. Tylczynski, Extended scattering continua characteristic of spin fractionalization in the two-dimensional frustrated quantum magnet Cs2CuCl4 observed by neutron scattering, Phys. Rev. B 68, 134424 (2003).
  52. D. A. Tennant, S. E. Nagler, D. Welz, G. Shirane, and K. Yamada, Effects of coupling between chains on the magnetic excitation spectrum of KCuF3, Phys. Rev. B 52, 13381 (1995).
  53. J. Lorenzana, G. Seibold, and R. Coldea, Sum rules and missing spectral weight in magnetic neutron scattering in the cuprates, Phys. Rev. B 72, 224511 (2005).
  54. A. V. Chubukov and D. I. Golosov, Quantum theory of an antiferromagnet on a triangular lattice in a magnetic field, J. Phys.: Condens. Matter 3, 69 (1991).
  55. C. Griset, S. Head, J. Alicea, and O. A. Starykh, Deformed triangular lattice antiferromagnets in a magnetic field: Role of spatial anisotropy and Dzyaloshinskii-Moriya interactions, Phys. Rev. B 84, 245108 (2011).

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Phys. Rev. B 112, 104413– Published 9 September, 2025
  • Received 9 May 2025
  • Revised 7 July 2025
  • Accepted 28 July 2025
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